DEMONSTRATION, AND NECESSARY TRUTHS. 297 



tions, and that those definitions are assumed to be 

 correct descriptions, as far as they go, of the objects 

 with which geometry is conversant. Now we have 

 pointed out that, from a definition as such, no propo- 

 sition, unless it be one concerning the meaning of a 

 word, can ever follow; and that what apparently 

 follows from a definition, follows in reality from an 

 implied assumption that there exists a real thing con- 

 formable thereto. This assumption, in the case of 

 the definitions of geometry, is false : there exist no 

 real things exactly conformable to the definitions. 

 There exist no points without magnitude; no lines 

 without breadth, nor perfectly straight ; no circles 

 with all their radii exactly equal, nor squares with 

 all their angles perfectly right. It will perhaps be 

 said that the assumption does not extend to the 

 actual, but only to the possible, existence of such 

 things. I answer that, according to any test we have 

 of possibility, they are not even possible. Their exist- 

 ence, so far as we can form any judgment, would 

 seem to be inconsistent with the physical constitution 

 of our planet at least, if not of the universe. To get 

 rid of this difficulty, and at the same time to save the 

 credit of the supposed systems of necessary truth, it 

 is customary to say that the points, lines, circles, and 

 squares which are the subject of geometry, exist in our 

 conceptions merely, and are part of our minds; which 

 minds, by working on their own materials, construct 

 an a priori science, the evidence of which is purely 

 mental, and has nothing whatever to do with outward 

 experience. By howsoever high authorities this doc- 

 trine may have been sanctioned, it appears to me 

 psychologically incorrect. The points, lines, circles, 

 and squares, which any one has in his mind, are (I 

 apprehend) simply copies of the points, lines, circles, 



